pure subgroup

Definition. A pure subgroup H of an abelian groupMathworldPlanetmath G is

  1. 1.

    a subgroupMathworldPlanetmathPlanetmath of G, such that

  2. 2.

    HmG=mH for all m.

The second condition says that for any hH such that h=ma for some integer m and some aG, then there exists bH such that h=mb. In other words, if h is divisible in G by an integer, then it is divisible in H by that same integer. Purity in abelian groups is a relative notion, and we denote H<pG to mean that H is a pure subgroup of G.

Examples. All groups mentioned below are abelian groups.

  1. 1.

    For any group, two trivial examples of pure subgroups are the trivial subgroup and the group itself.

  2. 2.

    Any divisible subgroup (http://planetmath.org/DivisibleGroup) or any direct summandMathworldPlanetmath of a group is pure.

  3. 3.

    The torsion subgroup (= the subgroup of all torsion elements) of any group is pure.

  4. 4.

    If K<pH, H<pG, then K<pG.

  5. 5.

    If H=i=1Hi with HiHi+1 and Hi<pG, then H<pG.

  6. 6.

    In Zn2, n is an example of a subgroup that is not pure.

  7. 7.

    In general, m<pZn if gcd(s,t)=1, where s=gcd(m,n) and t=n/s.

Remark. This definition can be generalized to modules over commutative rings.

Definition. Let R be a commutative ring and :0ABC0 a short exact sequenceMathworldPlanetmathPlanetmath of R-modules. Then is said to be pure if it remains exact after tensoring with any R-module. In other words, if D is any R-module, then


is exact.

Definition. Let N be a submoduleMathworldPlanetmath of M over a ring R. Then N is said to be a pure submodule of M if the exact sequencePlanetmathPlanetmathPlanetmathPlanetmath


is a pure exact sequence.

From this definition, it is clear that H is a pure subgroup of G iff H is a pure -submodule of G.

Remark. N is a pure submodule of M over R iff whenever a finite sum


where miM and riR implies that


for some niN. As a result, if I is an ideal of R, then the purity of N in M means that NIM=IN, which is a generalization of the second condition in the definition of a pure subgroup above.

Title pure subgroup
Canonical name PureSubgroup
Date of creation 2013-03-22 14:57:47
Last modified on 2013-03-22 14:57:47
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 20K27
Classification msc 13C13
Defines pure submodule
Defines pure exact sequence